A number of the form `8^(n)` , where n is a natural number greater than 1, cannot be divisible by __________.

Options

A. 1
B. 40
C. 64
D. `2^(2n)`

Answer

To solve the problem, we need to determine which of the given options cannot divide the number of the form \( 8^n \), where \( n \) is a natural number greater than 1.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The number \( 8^n \) can be rewritten using its prime factorization. Since \( 8 = 2^3 \), we have:
\[ 8^n = (2^3)^n = 2^{3n} \]

2. **Analyzing the Options**:
We need to check the divisibility of \( 8^n \) (or \( 2^{3n} \)) by each of the options provided:
- **Option 1: 1**
- Any number is divisible by 1. Thus, \( 8^n \) is divisible by 1.
- **Option 2: 40**
- The prime factorization of 40 is \( 2^3 \times 5 \). For \( 8^n \) to be divisible by 40, it must contain at least \( 2^3 \) and 5. Since \( 8^n = 2^{3n} \), it is divisible by \( 2^3 \) but does not contain the factor 5. Thus, \( 8^n \) is **not divisible** by 40.
- **Option 3: 64**
- The prime factorization of 64 is \( 2^6 \). Since \( 3n \) (from \( 8^n \)) is greater than or equal to 6 for \( n \geq 2 \), \( 8^n \) is divisible by 64.
- **Option 4: \( 2^{2n} \)**
- Since \( 3n \) is greater than \( 2n \) for \( n > 1 \), \( 8^n \) is also divisible by \( 2^{2n} \).

3. **Conclusion**:
The only option that \( 8^n \) is not divisible by is **40**.

### Final Answer:
The number of the form \( 8^n \) cannot be divisible by **40**.